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Report from “Workshop on Higher Gauge Theory”

Posted by Urs Schreiber

The two morning lectures were about classical solutions to equations of motion of the charged nn-particle, for n=1n=1 and n=2n=2.

First Iskander Taimanov reviewed a long list of (partially very old) results concerning the space of solutions of the Nambu-Goto particle coupled to an electromagnetic field, and propagating on manifolds of various sorts.

The method of choice here is to pick an arbitrary trajectory – one that doesn’t satisfy its equations of motions in that it doesn’t extremize the action functional, and then follow the gradient flow of the action functional, in the hope of reaching the physical solutions this way.

For instance one can estimate the size of the homology groups of the space of solutions this way, things like that.

In the second talk Dennis Koh from Potsdam talked about work on trying to generalize this to n=2n=2, the charged string: what can we say about its classical solutions given an arbitrary Kalb-Ramond-gerbe it couples too?

Again, the main idea was to look at the gradient flow. But now of course everything becomes more complicated. At the moment people can only show that the gradient flow exists for short times, but it may blow up later on.

The talk ended with mentioning that one might have to add other terms to the action functional (background gravity + background Kalb-Ramond) in order to regularize the flow.

My obvious guess would be: turn on the dilaton, just like Perelman did. Only difference is that in this case we are not interested in the flow on target space coming from the beta-functional equations induced by the action functional, but in the worldsheet action functional itself.

Prof. Huisken seemed to be interested in that idea…

I was surprised to learn in Taimanov’s talk how much there is to say about the space of classical solutions of the (electromagnetically!) charged 1-particle already, how deep the math involved is – and how huge the open questions still are.

Just imagine what will happen here when we allow 1-particles charged under gauge groups other than U(1)U(1), – and when we pass to n=2n=2, – and when we do both…

My impression was, with Huisken being in the audience, that with the recent interest in the completion of Hamilton’s approach to the Poincaré conjecture by gradient flow methods also these old “toy models” for gradient flow (namely flows along the gradients of ordinary action functionals of classical particles) are receiving renewed attention.

And what I find striking is that on the one hand we have these world-volume action functionals themselves, those of the string notably, whose gradient flow itself is interesting, since it knows about the classical trajectories of these objects, while on the other hand from these very action functionals one obtains those beta-functional renormalization group flow equations, which are themselves a gradient flow, now of what is called in physics the “background field action”.

Of course from a physical standpoint this is a banality, at least since the discovery of gravity in string theory, but now in light of the fact of the mathematical interest that these gradient flows – on the worldsheet as well as on the corresponding target space – receive, this seems to be rather remarkable, once again.

Well, I have quite generally the impression that with the fast-forward progress of the string theory investigation apparently having slowed down a bit as of late, there are lots of bits and pieces that have long been laid ad acta by hep physicists and which still are awaiting their full investigation and development to sound mathematical structures in their own right.

Okay, enough rambling, On with the report.

Sections and Fibre Integration in terms of Deligne Cohomology

Christian Becker talked about work trying to clarify the notion of

a) sections

and of

b) fiber integration

for abelian nn-gerbes with connection.

The work reported on is entirely based on using the Deligne cocycle description of abelian nn-gerbes.

What is a “section” of such a Deligne class? I believe that the definition that Christian Becker presented is precisely what one gets when one takes the general definition of section of an (n+1)(n+1)-bundle with connection and then restricts everything in sight to be abelian and finally passing to local trivializations and descent data.

So they find the section of an abelian gerbe to exist if and olny if that gerbe is trivializable (globally) in which case the section is the trivializing line bundle.

The definition Becker presented involved pulling back the Deligne class in the obvious way and then simply integrating all forms that appear in the cocycle separetely. This can’t be the general prescription, and in fact it isn’t. But it does work in simple special cases, some of them useful in applications.

In private discussion with others later on I dared to mention that various open questions here do have natural answers, but met with quite some resistance regarding the fact that this involves higher categories. Some are trying to fight it.

I am not sure yet if I follow the claim that the fusion product can be understood “geometrically” only using gerbe bi-modules. After all, ordinary gerbe modules carry a canonical fusion product just by themselves.

Moreover, in the degenerate case where the gerbe is trivial, we know that the product on its “modules” (which then are just ordinary vector bundles) is nothing but the ordinary product in ordinary (untwisted) K-theory, namely just tensor product of vector bundles.

On the other hand, after the dust has settled one finds that these bibranes that are related to the fusion product do crucially involve the space of flat connections on the 3-holed sphere. And that’s of course precisely the structure which is known to govern the fusion product of these twisted groupoid reps…

One nice application of this bibrane formalism might be, I am thinking, that it seems to provide a way to consistently formulate the sigma-model for the string propagating in “T-folds”, since it allows to formulate the WZW term in cases where parts of the worldsheet map into different, but T-dual, spacetimes.

A “T-fold” is like a manifold, only that one allows for gluing of patches not just diffeomorphism, but also T-duality transformations if both patches can be written as torus bundles. See for instance

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Re: Report from “Workshop on Higher Gauge Theory”

I am not sure yet if I follow the claim that the fusion product can be understood “geometrically” only using gerbe bi-modules.

I guess that’s just because one cannot in general fuse (left-)modules, or did I misunderstand what you were referring to? You can in the Cardy case, but that’s accidental.
Bimodules, on the other hand, will always do.

Re: Report from “Workshop on Higher Gauge Theory”

Not if by “fuse” you already mean tensoring by coequalizing the joint action.

But one can certainly have other tensor products on just one-sided modules.

But I guess what is going on here is actually that the non-bimodule way of saying this can just equivalently be reformulated in terms of ordinary composition of certain bimodules. Certainly both constructions involve the same ingredients here, like a correspondence space (trinion/figure-eight) and the space of flat GG-connections on it. So it’s probably just a matter of perspective.

Re: Report from “Workshop on Higher Gauge Theory”

In private discussion with others later on I dared to mention to that various open questions here do have natural answers, but met with quite some resistance regarding the fact that this involves higher categories. Some are trying to fight it.

Re: Report from “Workshop on Higher Gauge Theory”

quote: “Many people have the impression, that higher category theory is getting nowhere.”

I believe that one reason for that is that higher categories is too powerful. When the problem is solved too effortlessly and thereby generalized too much, the usual feeling that you get after applying a gimlet to a thick board for a while doesn’t occur. So the usual “I have solved it”-feeling isn’t there.

For instance the authors of this D-brane paper make a big deal of a certain diagrammatic notation which they introduce. But really it’s nothing but a tiny special case of string diagram notation for 2-categories.

Note: this is not an example for disdain – none of these authors has expressed to me any disdain for category theory, rather one has done the opposite. But the point is (which I know from personal experience) that when one turns this around and starts by saying that one applies 2-categories to D-brane theory, people will try to either walk away or say “No, I don’t want to hear that.”

Similar effects arise when lots of tedious work using Deligne cocycles is claimed to have an easy interpretation once one draws a handful of arrows. That’s too easy to be good.

Re: Report from “Workshop on Higher Gauge Theory”

I see. Nasty. There’s nothing else to be done but to try and make connections with as much maths and physics as possible. Let it never be said of our works, “The authors of the Gruppenpest wrote papers which were incomprehensible to those like me who had not studied group theory”. We must make every attempt to allow our works to be understood by all comers, especially to those who are “hard-sells”.

Re: Report from “Workshop on Higher Gauge Theory”

There are probably only two ways to make people care about n-categories in the short term:
1. Use n-categories to predict some definite experimental signature at the LHC, which is then experimentally confirmed. You have a golden window of about two years to do that.
2. Convince Ed Witten that n-categories are important (the Green-Schwarz strategy).

Unless you succeed with any of those, you have to resort to the terrier strategy. Just keep doing your thing despite the headwind, and explain why it must be important. However, expect this to take a long time, perhaps much longer than your grants will last, and in the meantime expect ridicule by those who have never had an original thought themselves (“the mainstream”). Fifteen years after the discovery of the multi-dimensional Virasoro algebra, and more than 20 years after the multi-dimensional affine algebra, it is still no big hit, despite the simple proof that the natural completion of the gauge algebra is incompatible with nonzero charge without an anomaly.

Re: Report from “Workshop on Higher Gauge Theory”

I can seen that physicists will be harder to crack. How about mathematicians? We have Yuri Manin with us:

…what happens is the slow emergence of the following hierarchical
picture. Categories themselves form objects of a larger category Cat morphisms in which are functors, or “natural constructions” like a (co)homology theory of topological spaces. However, functors do not form simply a set or a class: they also form objects of a category. Axiomatizing this situation we get a notion of 2-category whose prototype is Cat. Treating 2-categories in the same way, we get 3-categories etc.

The following view of mathematical objects is encoded in this hierarchy: there is no equality of mathematical objects, only equivalences. And since an equivalence is also a mathematical object, there is no equality between them, only the next order equivalence etc., ad infinitim.

This vision, due initially to Grothendieck, extends the boundaries of classical mathematics, especially algebraic geometry, and exactly in those developments where it interacts with modern theoretical physics.

Re: Report from “Workshop on Higher Gauge Theory”

This was very amusing!

The degree to which this is amusing depends a little on how much one’s daily bread depends on the opinion of others on such matters, unfortunately.

I am certainly a little worried that just uttering the words “higher category” has a high potential of inducing a disdain among some (and in this case among those who already ask precisely the right questions whose only natural answer does involve these words) which is comparable only to that which the words “string theory” invoke in others.

(Now this gives an interesting classification of all people in terms of the two parameters “disgust for category theory” and “digust for string theory”. )

I imagine it’s similar to the Gruppenpest-phenomenon of ancient times – and indeed, googling for that of course leads to John Baez making exactly that comparison.

Re: Report from “Workshop on Higher Gauge Theory”

Uh oh. You have uttered the words I didn’t think I would ever here you utter. “Daily bread”.

If you ever become concerned about daily bread, let me entice you to the dark side. Remember the guy who wrote you about possibly using higher categories in finance? He and I are now co-workers not far from Riverside. We’d love to have you join us *muahahaha* :)

The dynamics of stock prices are akin to that of a point particle and the dynamics of interest rates are akin to open strings.

Annals of Science
Crash Course
by Elizabeth Kolbert
(page 7)
The New Yorker
9 May 2007

“… Unless funding for another collider materializes, a lot of experimentalists will soon find themselves out of work. “Half of those guys already have résumés in at hedge funds,” one theorist joked to me. Arguably, the theorists’ situation is not all that much more secure; at a certain point, speculations about the nature of the universe that can’t be put to the test cease to be physics….”

The Price of Truth
How Money Affects the Norms of Science
by David B. Resnick
Oxford University Press, 2006
Review by Tony O’Brien, RN, MPhil on May 1st 2007
Volume: 11, Number: 18
http://mentalhelp.net/books/books.php?type=de&id=3616

The theme of The Price of Truth is that the ideal of science as the objective, disinterested pursuit of knowledge is just that, an ideal, and that modern science is intimately tied up with the business world, and with financial incentives of one sort or another. While there are some who would see this state of affairs as a travesty, Resnik is more pragmatic. Drawing on examples of classical scientists, and from the current practice of science, Resnik argues for a middle road, one in which there can be room for financial incentives to encourage science, but where there are adequate restraints on the excesses of money to maintain the more communitarian goals of science.